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Statistics, Causality and Bell's Theorem

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Statistics, Causality and Bell's Theorem Statistical Science 2014, Vol. 29, No. 4, 512–528 DOI: 10.1214/14-STS490 c Institute of Mathematical Statistics, 2014 Statistics, Causality and Bell’s Theorem Richard D. Gill Abstract. Bell’s [Physics 1 (1964) 195–200] theorem is popularly sup- posed to establish the nonlocality of quantum physics. Violation of Bell’s inequality in experiments such as that of Aspect, Dalibard and Roger [Phys. Rev. Lett. 49 (1982) 1804–1807] provides empirical proof of nonlocality in the real world. This paper reviews recent work on Bell’s theorem, linking it to issues in causality as understood by statis- ticians. The paper starts with a proof of a strong, finite sample, version of Bell’s inequality and thereby also of Bell’s theorem, which states that quantum theory is incompatible with the conjunction of three formerly uncontroversial physical principles, here referred to as locality, realism and freedom. Locality is the principle that the direction of causality matches the direction of time, and that causal influences need time to propagate spatially. Realism and freedom are directly connected to statistical thinking on causality: they relate to counterfactual reasoning, and to randomisation, respectively. Experimental loopholes in state-of-the-art Bell type experiments are related to statistical issues of post-selection in observational studies, and the missing at random assumption. They can be avoided by properly matching the statistical analysis to the ac- tual experimental design, instead of by making untestable assumptions of independence between observed and unobserved variables. Method- ological and statistical issues in the design of quantum Randi challenges (QRC) are discussed. The paper argues that Bell’s theorem (and its experimental confir- mation) should lead us to relinquish not locality, but realism. Key words and phrases: Counterfactuals, Bell inequality, CHSH in- equality, Tsirelson inequality, Bell’s theorem, Bell experiment, Bell test loophole, nonlocality, local hidden variables, quantum Randi challenge. arXiv:1207.5103v6 [stat.AP] 30 Jan 2015 1. INTRODUCTION the conjunction of three fundamental principles of classical physics which are sometimes given the short Bell’s (1964) theorem states that certain predic- tions of quantum mechanics are incompatible with names “realism”, “locality” and “freedom”. Corre- sponding real world experiments, Bell experiments, are supposed to demonstrate that this incompati- Richard D. Gill is Professor, Mathematical Institute, University of Leiden, Niels Bohrweg 1, Leiden, 2333 bility is a property not just of the theory of quan- CA, The Netherlands e-mail: tum mechanics, but also of nature itself. The conse- [email protected]; URL: quence is that we are forced to reject at least one of http://www.math.leidenuniv.nl/˜gill. these three principles. This is an electronic reprint of the original article Both theorem and experiment hinge around an published by the Institute of Mathematical Statistics in inequality constraining probability distributions of Statistical Science, 2014, Vol. 29, No. 4, 512–528. This outcomes of measurements on spatially separated reprint differs from the original in pagination and physical systems; an inequality which must hold if typographic detail. all three fundamental principles are true. In a nut- 1 2 R. D. GILL shell, the inequality is an empirically verifiable con- experiments which are not performed; and even if sequence of the idea that the outcome of one mea- we do that, why should we demand that these coun- surement on one system cannot depend on which terfactual objects satisfy the same kind of physical measurement is performed on the other. This idea, constraints as the factual objects? However, it is a called locality or, more precisely, relativistic local fact that prior to quantum physics, realism was a causality, is just one of the three principles. Its completely natural property of all physical theories. formulation refers to outcomes of measurements The concepts of realism and locality together are which are not actually performed, so we have to often considered as one principle called local realism. assume their existence, alongside of the outcomes Local realism is implied by the existence of local hid- of those actually performed: the principle of real- den variables, whether deterministic or stochastic. ism, or more precisely, counterfactual definiteness. In a precise mathematical sense, the reverse impli- Finally, we need to assume that we have complete cation is also true: local realism implies that we can freedom to choose which of several measurements to construct a local hidden variable (LHV) model for perform—this is the third principle, also called the the phenomenon under study. However, one likes to no-conspiracy principle or no super-determinism. think of this assumption (or pair of assumptions), (As we shall see, super-determinism is a conspira- the important thing to realize is that it is a com- torial form of determinism.) pletely unproblematic feature of all classical physi- We shall implement the freedom assumption as cal theories; freedom (no conspiracy) even more so. the assumption of statistical independence between The connection between Bell’s theorem and statis- the randomisation in a randomised experimental de- tical notions of causality has been noted many times sign, and the set of outcomes of each experimen- tal unit under all possible treatments. This set con- in the past. For instance, in a short note, Robins, sists of the “counterfactual” outcomes of those treat- VanderWeele and Gill (2015) derive Bell’s inequal- ments which were not actually applied, as well as ity using the statistical language of causal interac- the “factual” outcome belonging to the treatment tions. The causal graph (DAG) of observed and un- chosen by the randomisation. observed variables corresponding to a classical phys- By existence of the outcomes of not actually per- ical description of one run of a standard Bell exper- formed experiments, we mean their mathematical iment is given in Figure 1. Alice and Bob’s settings existence within some mathematical-physical theory are binary: they independently use randomisation (a of the phenomenon in question. So “realism” actu- coin toss) to choose between one of two settings on a ally refers to models of reality, not to reality itself. measurement device. The outcome of each measure- Moreover, it could be thought of as a somewhat ment is also binary. Observed variables are repre- idealistic position. If we already have an adequate sented by grey rectangles; unobserved (there is only mathematical physical model of reality, there would one, but of course it might be of arbitrarily complex not seem to be a pressing need to add into this the- nature) by a white oval. The validity of this causal ory some mathematical description of outcomes of model places restrictions on the joint distribution of Fig. 1. A classical description of a Bell-CHSH type experiment entails the validity of the graphical model described by this simple causal graph. Rectangles: observed variables; ellipse: unobserved. Settings and outcomes are both binary. Experimental results arguably (via Bell’s theorem) show that the classical description has to be abandoned. The probability distribution of experimental data is far outside the class of probability distributions allowed by the model. STATISTICS, CAUSALITY AND BELL’S THEOREM 3 the observed variables; see, for instance, Ver Steeg get to see the value of A or A′, and either the value and Galstyan (2011). of B or B′. We can therefore determine the value In view of the experimental support for violation of just one of the four products AB, AB′, A′B, and ′ ′ 1 of Bell’s inequality, the present writer prefers to A B , each with equal probability 4 , for each row imagine a world in which “realism” is not a fun- of the table. Denote by AB obs the average of the damental principle of physics but only an emergent observed products of A hand iB (“undefined” if the ′ ′ property in the familiar realm of daily life. In this sample size is zero). Define AB obs, A B obs and way, we can keep quantum mechanics, locality and A′B′ similarly. h i h i h iobs freedom. This position does entail taking quantum ′ ′ randomness very seriously: it becomes an irreducible Fact 1. For any four numbers A, A , B, B each equal to 1, feature of the physical world, a “primitive notion”; ± ′ ′ ′ ′ it is not “merely” an emergent feature. He believes (1) AB + AB + A B A B = 2. that within this position, the measurement problem − ± (Schr¨odinger cat problem) has a decent mathemat- Proof. Notice that ical solution, in which causality is the guiding prin- ′ ′ ′ ′ ′ ′ ′ AB + AB + A B A B = A(B + B )+ A (B B ). ciple (Slava Belavkin’s “eventum mechanics”). − − Many practical minded physicists claim to be ad- B and B′ are either equal to one another or unequal. herents of the so-called Many Worlds interpretation In the former case, B B′ =0 and B + B′ = 2; in (MWI) of quantum mechanics. In the writer’s opin- the latter case B B′−= 2 and B + B′ = 0.± Thus, ion (but also of many writers on quantum founda- AB + AB′ + A′B −A′B′ equals± either A or A′, both tions), this interpretation also entails a rejection of of which equal −1, times 2. All possibilities lead “realism”, but now in a very strong sense: the reality to AB + AB′ +±A′B A′B±′ = 2. of an actual random path taken by Nature through − ± space–time is denied. The only reality is the ensem- Fact 2. ble of all possible paths. Devilish experiments lead ′ ′ ′ ′ (2) AB + AB + A B A B 2. to dead cats turning up on some paths, and alive h i h i h i − h i≤ cats on others. According to MWI, the only real- Proof. By (1), ity is the quantum wave-function.
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